Maximal L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-regularity of the heat equation and application to a free boundary problem of the Navier-Stokes equations near the half-space

This is a survey of recent results concerning on maximal L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-regularity of the heat equation with the Naumann boundary condition in the half Euclidian space Ogawa and Shimizu (Proc Jpn Acad A, 96:57–62, 2020). It also includes maximal L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-regularity for the initial boundary value of the Stokes system in the half-space under the stress free boundary condition. As an application, we introduce the time global well-posedness for the free boundary problem of the incompressible Navier-Stokes equations under the small initial data in the half Euclidean spaces R+n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n_+$$\end{document} developed in Danchin-Hieber-Mucha-Tolksdorf (arXiv:2011.07918) and Ogawa and Shimizu (2021).